Sufficient conditions for smoothing codimension one foliations
Author:
Christopher Ennis
Journal:
Trans. Amer. Math. Soc. 276 (1983), 311322
MSC:
Primary 57R30; Secondary 57R10, 58F18
MathSciNet review:
684511
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Abstract: Let be a compact manifold. Let be a nonsingular vector field on , having unique integral curves through . For continuous, call whenever defined. Similarly, call . For , a foliation of is said to be smoothable if there exist a foliation , which approximates , and a homeomorphism such that takes leaves of onto leaves of . Definition. A transversely oriented Lyapunov foliation is a pair consisting of a codimension one foliation of and a nonsingular, uniquely integrable vector field on , such that there is a covering of by neighborhoods , , on which is described as level sets of continuous functions for which is continuous and strictly positive. We prove the following theorems. Theorem 1. Every transversely oriented Lyapunov foliation is smoothable to a transversely oriented Lyapunov foliation . Theorem 2. If is a transversely oriented Lyapunov foliation, with and continuous for and , then is smoothable to a transversely oriented Lyapunov foliation . The proofs of the above theorems depend on a fairly deep result in analysis due to F. Wesley Wilson, Jr. With only elementary arguments we obtain the version of Theorem 1. Theorem 3. If is a transversely oriented Lyapunov foliation, with and is continuous, then is smoothable to a transversely oriented Lyapunov foliation .
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 A. Denjoy, Sur les courbes définies par les équations differentielles à la surface du tore, J. Math. Pures Appl. 11 (1932), 333375.
 [2]
 C. Ennis, M. Hirsch and C. Pugh, Foliations that are not approximable by smoother ones, Report PAM63, Center for Pure and Appl. Math., University of California, Berkeley, Calif., 1981.
 [3]
 J. Harrison, Unsmoothable diffeomorphisms, Ann. of Math. (2) 102 (1975), 8594. MR 0388458 (52:9294)
 [4]
 D. Hart, On the smoothness of generators for flows and foliations, Ph.D. Thesis, University of California, Berkeley, Calif., 1980.
 [5]
 A. J. Schwartz, A generalization of a PoincaréBendixson theorem to closed twodimensional manifolds, Amer. J. Math. 85 (1963), 453458. MR 0155061 (27:5003)
 [6]
 D. Sullivan, Hyperbolic geometry and homeomorphisms, Geometric Topology (J. Cantrell, editor), Academic Press, New York, 1979, p. 549. MR 537749 (81m:57012)
 [7]
 F. W. Wilson, Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc. 139 (1969), 413428. MR 0251747 (40:4974)
 [8]
 , Implicit submanifolds, J. Math. Mech. 18 (1968), 229236. MR 0229252 (37:4826)
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DOI:
http://dx.doi.org/10.1090/S00029947198306845114
PII:
S 00029947(1983)06845114
Article copyright:
© Copyright 1983
American Mathematical Society
